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Properties

Label 4-3744e2-1.1-c0e2-0-2
Degree $4$
Conductor $14017536$
Sign $1$
Analytic cond. $3.49129$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·25-s + 2·37-s + 2·41-s + 4·53-s − 4·61-s + 2·73-s + 2·89-s + 2·97-s + 2·109-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s − 4·185-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·5-s + 2·25-s + 2·37-s + 2·41-s + 4·53-s − 4·61-s + 2·73-s + 2·89-s + 2·97-s + 2·109-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s − 4·185-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14017536\)    =    \(2^{10} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.49129\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14017536,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9184381622\)
\(L(\frac12)\) \(\approx\) \(0.9184381622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$ \( ( 1 - T )^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.881280032712771804404653911545, −8.518609341742377314456769295243, −7.905175551103517088204351829065, −7.65588509238764072186343001743, −7.60879150908012125039689572300, −7.33639717729686699128713140326, −6.60298547932210727145245960736, −6.41919343677687892111192996719, −5.86087061692939696009033054717, −5.62912378633963076556557223588, −4.82838919246430270098992968157, −4.70647373636620488158144824503, −4.21580925294549505121109840891, −3.89430784817368792590338407325, −3.57539908031064634315522150312, −3.06771659470846232960731048340, −2.50902055084983847112073041410, −2.17349867152280524075897364071, −1.10247418661650765586022074307, −0.66156577215943567323766281094, 0.66156577215943567323766281094, 1.10247418661650765586022074307, 2.17349867152280524075897364071, 2.50902055084983847112073041410, 3.06771659470846232960731048340, 3.57539908031064634315522150312, 3.89430784817368792590338407325, 4.21580925294549505121109840891, 4.70647373636620488158144824503, 4.82838919246430270098992968157, 5.62912378633963076556557223588, 5.86087061692939696009033054717, 6.41919343677687892111192996719, 6.60298547932210727145245960736, 7.33639717729686699128713140326, 7.60879150908012125039689572300, 7.65588509238764072186343001743, 7.905175551103517088204351829065, 8.518609341742377314456769295243, 8.881280032712771804404653911545

Graph of the $Z$-function along the critical line